331 



In suiranary it appears plausible to assume that 

 the basic picture of pressure fluctuations arising 

 from mean-shear turbulence interactions will be 

 unchanged in a Lagrangian frame of reference, 

 although the actual spectra are different. The 

 postulated relations for Lagrangian spectra should 

 be directly applicable to any Lagrangian phenomenon; 

 in particular the relations should be applicable 

 to the inception of nuclei in a fluctuating pressure 

 field. 



In relating the information on the pressure field 

 to the problem at hand, it is evident that two 

 criteria must be satisfied for turbulence induced 

 inception: 



1) The pressure must dip to the vapor pressure 

 or lower. 



2) The pressure minimum must persist for a time 

 that is long in comparison to the characteristic 

 time scale of the bubble, say Tg (taken to be the 

 time scale for growth at inception) . 



Both factors lead to scale effects. Consider 

 first the second factor. The preceding arguments 

 for the pressure field in a Lagrangian frame of 

 reference lead to the hypothetical spectrum shown 

 in Figure 2. For convenience we have normalized 

 the spectrum with respect to the mean square pressure 

 and the Lagrangian time scale "J . (c.f. Tennekes 

 and Lumley, Chapter 8). Requirement (2) for bubble 

 growth is plotted at the frequency oi = 1/T . It 

 is clear that as long as u << l/Tg, any pressure 

 flucuation persists for a time longer than the 

 time scale of the bubble. Thus at frequencies less 

 than w = 1/Tg cavitation inception can occur with 

 minimal local tension. Moreover, by integrating 

 the spectrum from u = to co = 1/T , we can deter- 

 mine that fraction of the mean square pressure 

 which can contribute to bubble growth without 

 appreciable tension (assuming a normal distribution 

 of nuclei) . 



Consider now the effect of maintaining Tg con- 

 stant while varying the Reynolds number. Taking 

 V ~ 5,/u' and noting that there are essentially no 

 pressure fluctuations of interest above the 

 Kolmogorov frequency, u = (e/v)'2 we find that 

 after 1/Tg, exceeds (e/v)'2, the entire spectrum 

 can potentially contribute to bubble growth. This 

 will occur when the Reynolds number is roughly 



Mean Square Pressure Fluctuation (Lagrangian) 



on%)\ 



FIGURE 3. Integration of Lagrangian pressure 

 spectrum. 



uJl/v ~ (H/uTg)^. By noting the spectral dependence 

 on frequency and performing a running integral, a 

 plot such as shown in Figure 3 can be generated. 

 This graph illustrates how rapidly the asymptotic 

 state is reached. This occurs when ^/T -^> ::' (e/V)'^ 

 > (u'll/V)'5 or when )i/u'T > (u'il/V)'5 as previously 

 stated. 



As an example,* cavitation is observed to occur 

 in siibmerged jets at an axial position, x, that is 

 roughly one diameter from tihe nozzle. Assuming 

 the dissipation rate to be approximately O.OSUj /x, 

 where Uj is the jet velocity, results in a criterion 

 that the jet diameter must exceed the following 



before scale effects are absent: d > O.OSUi 



Vv. 



FIGURE 2. Hypothetical pressure spectrum in a 

 Lagrangian frame of reference. 



Using typical values of Uj = 10 m/s and Tg = 10 

 sec, we conclude that the asymptote is reached for 

 d ~ 50 meters. Thus size effects could be important 

 in many model experiments. 



Effect of Intermittency at Small Scale 



In 1947, Batchelor and Townsend concluded from 

 observations of the velocity derivatives in 

 turbulent flow that tlie fine structure of the 

 turbulence (small scales, high frequency) was 

 spatially localized and highly intermittant in 

 high Reynolds number flows. Subsequent work [c.f. 

 Kuo and Corrsin (1971)] has confirmed that there 

 is a decrease in the relative volume occupied by 

 the fine structure as the Reynolds number is 

 increased. Thus tlie spatial intermittancy increases 

 with Reynolds number. The effect of this phenomenon 

 on filtered hot wire signals is shown in Figure 4. 

 These data are derived from Kuo and Corrsin (1971) . 

 It is obvious from these data tJiat the signal is 

 increasingly intermittant as the filter frequency 

 is moved to higher and higher values . 



Since the dissipation of turbulent energy takes 

 place at the smallest scales of motion, it is clear 

 from these observations that the rate of dissipation 

 of turbulent energy must vary widely with space 

 and time. It was this consideration that led 



*Strictly speaking, these results are only applicable 

 when the Lagrangian turbulent field is stationary. 

 In most flows of interest this is seldom the case. 

 However, the smallest scales of motion can often 

 be considered to be in quasl-equilibrium. 



